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##

Related Eigenproblems

- If and are Hermitian, is not positive definite, but
is positive definite for some choice of real numbers
and , one can solve the generalized Hermitian
eigenproblem
instead.
Let
;
then the eigenvectors of and
are identical.
The eigenvalues of and
the eigenvalues
of
are related by
.

- If and are non-Hermitian, but
and
are Hermitian, with positive definite,
for easily determined , and nonsingular and ,
then one can compute the eigenvalues and eigenvectors
of
.
One can convert these to eigenvalues and eigenvectors of via
and .
For example, if is Hermitian positive definite but is skew-Hermitian
(i.e., ), then is Hermitian, so we may choose
, , and .
See §2.5
for further discussion.

- If one has the GHEP
,
where and are Hermitian and is positive definite, then
it can be converted to a HEP as follows.
First, factor , where is any nonsingular matrix (this is
typically done using Cholesky factorization). Then solve the
HEP for
. The eigenvalues of
and are identical, and if is an
eigenvector of , then
satisfies
.
Indeed, this is a standard algorithm for .

- If and are positive definite with
and for some rectangular matrices and ,
then the eigenproblem for
is equivalent to the
*quotient singular value decomposition (QSVD)*
of and , discussed in §2.4.
The state of algorithms is such that it is probably better to try solving
the eigenproblem for than computing the QSVD of and .

** Next:** Example
** Up:** Generalized Hermitian Eigenproblems
** Previous:** Specifying an Eigenproblem
** Contents**
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Susan Blackford
2000-11-20